suppose that tan(t)=12/5, where t is in the third quadrant. Find tan(2t)

We can use this identity :

tan (2t)   =  [2 tan ( t) ] /  [ 1 - tan^2 (t) ]  =

2 (12/5)  /  [ 1  - (12/5)^2 ]  =

(24 / 5)   / [ 1 -   144 / 25 ]  =

(24 / 5)   /  (  [25 - 144] / 25  )  = 

(24 / 5)  /  (  -119/25 )  =

(24 / 5)  *  ( -25 / 119)   =

(-24 * 5) / 119 =

-120 / 119